How were you planning on accessing Sky Club? Do you have an Amex Plat card or DL Amex Reserve card? If you only have DL STE+ status, then this is the section that applies and you need to be at least traveling in Delta PS cabin or equivalent PE cabin on partners for access. If you have a DL Amex Reserve card, you must be flying on either DL operated or a DL marketed flight. Since you are flying on a KL marketed and operated flight, you won't have comp access with an DL Amex Reserve card (you can still potentially access with DL Amex Reserve, but there is a $50 fee -- assuming they can't actually see you are on a Light fare).

https://www.delta.com/us/en/delta-sky-club/access






This is the relevant section for DL Amex Reserve cardmembers